MathDB

2

Part of 1992 National High School Mathematics League

Problems(2)

Part of Unit Circle

Source: 1992 National High School Mathematics League, Exam One, Problem 2

2/26/2020
The equation of unit circle in Quadrant I, III, IV ((−1,0),(1,0),(0,−1),(0,1)(-1,0),(1,0),(0,-1),(0,1) included) is (A)(x+1−y2)(y+1−x2)=0\text{(A)}(x+\sqrt{1-y^2})(y+\sqrt{1-x^2})=0 (B)(x−1−y2)(y−1−x2)=0\text{(B)}(x-\sqrt{1-y^2})(y-\sqrt{1-x^2})=0 (C)(x+1−y2)(y−1−x2)=0\text{(C)}(x+\sqrt{1-y^2})(y-\sqrt{1-x^2})=0 (D)(x−1−y2)(y+1−x2)=0\text{(D)}(x-\sqrt{1-y^2})(y+\sqrt{1-x^2})=0
Capacity

Source: 1992 National High School Mathematics League, Exam Two, Problem 2

2/27/2020
Define set Sn={1,2,⋯ ,n}S_n=\{1,2,\cdots,n\}. XX is a subset of SnS_n. We call sum of all numbers in XX capacity (capacity of empty set is 00). If capacity of XX is odd/even, then we call it odd/even subset. (a) Prove that the number of odd subsets and even subsets of SnS_n are the same. (b) Prove that the sum of capacity of all odd subsets and even subsets are the same when n≥3n\geq3. (c) Calculate the sum of capacity of all odd subsets when n≥3n\geq3.